Optimal. Leaf size=42 \[ \frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sin (a+b x) \sqrt {\cos (a+b x)}}{3 b} \]
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Rubi [A] time = 0.06, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {3310} \[ \frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sin (a+b x) \sqrt {\cos (a+b x)}}{3 b} \]
Antiderivative was successfully verified.
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Rule 3310
Rubi steps
\begin {align*} \int \left (-\frac {x}{3 \sqrt {\cos (a+b x)}}+x \cos ^{\frac {3}{2}}(a+b x)\right ) \, dx &=-\left (\frac {1}{3} \int \frac {x}{\sqrt {\cos (a+b x)}} \, dx\right )+\int x \cos ^{\frac {3}{2}}(a+b x) \, dx\\ &=\frac {4 \cos ^{\frac {3}{2}}(a+b x)}{9 b^2}+\frac {2 x \sqrt {\cos (a+b x)} \sin (a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.43, size = 40, normalized size = 0.95 \[ \frac {\sqrt {\cos (a+b x)} \left (4 x \sin (a+b x)+\frac {8 \cos (a+b x)}{3 b}\right )}{6 b} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos \left (b x + a\right )^{\frac {3}{2}} - \frac {x}{3 \, \sqrt {\cos \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int x \left (\cos ^{\frac {3}{2}}\left (b x +a \right )\right )-\frac {x}{3 \sqrt {\cos \left (b x +a \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \cos \left (b x + a\right )^{\frac {3}{2}} - \frac {x}{3 \, \sqrt {\cos \left (b x + a\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x\,{\cos \left (a+b\,x\right )}^{3/2}-\frac {x}{3\,\sqrt {\cos \left (a+b\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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